In economic management, we often use some (divisible) private resources to cooperative. Fuzzy coalitions always be used to describe this situation in cooperative fuzzy games. In this paper, we proposed two new solution concepts in NTU cooperative fuzzy games, and discussed their properties

Xu, Lijun

Yang, Wenbo

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Association for Information SystemsAIS Electronic Library (AISeL)WHICEB 2013 Proceedings Wuhan International Conference on e-BusinessSummer 5-25-2013A Situation of Economic Management in NTUCooperative Fuzzy GamesLijun XuCollege of Business Administration, Shenyang University, P. R. ChinaWenbo YangSchool of Economics, Fudan University, P. R. ChinaFollow this and additional works at: http://aisel.aisnet.org/whiceb2013This material is brought to you by the Wuhan International Conference on e-Business at AIS Electronic Library (AISeL). It has been accepted forinclusion in WHICEB 2013 Proceedings by an authorized administrator of AIS Electronic Library (AISeL). For more information, please contactelibrary@aisnet.org.Recommended CitationXu, Lijun and Yang, Wenbo, "A Situation of Economic Management in NTU Cooperative Fuzzy Games" (2013). WHICEB 2013Proceedings. 1.http://aisel.aisnet.org/whiceb2013/1The Twelfth Wuhan International Conference on E-Business——Emerging Operations & Services Management Track 833A Situation of Economic Management in NTU Cooperative Fuzzy GamesLijun Xu1, Wenbo Yang21College of Business Administration, Shenyang University, P. R. China2School of Economics, Fudan University, P. R. ChinaAbstract: In economic management, we often use some (divisible) private resources to cooperative. Fuzzycoalitions always be used to describe this situation in cooperative fuzzy games. In this paper, we proposed twonew solution concepts in NTU cooperative fuzzy games, and discussed their properties.Keywords: Bargaining set; NTU; Cooperative Fuzzy game1. INTRODUCTIONIn economic management, businessmen often possess some (divisible) private resources such ascommodities, time, money, and have to decide about the amount to be invested in the joint project. Cooperativegames with fuzzy coalitions are introduced in [1] and [2] to model the above situations where businessmen havethe possibility to cooperate with different participation levels, varying from non-cooperation to full cooperation,and where the obtained reward depends on the levels of participation.In this paper, we proposed more reasonable concepts of the bargaining sets which are very importantsolution concepts in game theory underlying NTU cooperative fuzzy game, and prove the inclusion relationbetween them. It is necessary to claim that the results of this paper were based on the work of Tijs[3] Muto[4] andZhang Fengrong[5] .2. PRELIMINARIESLet N={1,2,…,n} be a finite player set, and a fuzzy coalition is a vector s∈[0,1]N. The i-th coordinate siof sis the participation level of player i in the fuzzy coalition s. Instead of [0,1]N we will also write ΦN for the set offuzzy coalitions on player set N.For s∈ΦN we define the carrier of s by car(s)={i∈N| si>0} and call s a proper fuzzy coalition if car(s)≠N.The set of proper fuzzy coalitions on player set N is denoted by ΠΦN, and the set of non-empty proper fuzzycoalitions on player set N by ΠΦ0N. For s, t∈ΦN we use the notation s≤t expresses si ≤ ti for each i∈N. Wedefine s∧t=(min(s1, t1),…, min(sn, tn)) and s∨t=(max(s1, t1),…, max(sn, tn)).Let Rcar(s) indicates |car(s)|-dimension Euclid space. Denoted x≤y means xi≤yifor each i∈car(s); and x<<ymeans xi< yifor each i∈car(s) for any x, y∈Rcar(s).An NTU fuzzy cooperative game is a pair (N, V), where N={1,2,…,n} is a non-empty finite set, and V, thecharacteristic function, assigns to every fuzzy coalition s a V(s) which is a subset of Rcar(s), and satisfies thefollow properties:(1) V(0)=∅, where 0=(0,0,…,0).834 The Twelfth Wuhan International Conference on E-Business——Emerging Operations & Services Management Track(2) For each s∈ΦN, V(s) is a non-empty subset of Rcar(s), and it is closed and comprehensiveness. (that is tosay, if x∈V(s), y∈Rcar(s), and y ≤ x, then y∈V(s), too).(3) For each x∈Rcar(s), V(s)∩(x+R+car(s)) is bounded.(4) V(s) is nonlevelness, i.e., if x,y∈V0(s), and x≥y, then x=y, where, V0(s)={x∈V(s)|there is no y∈V(s), suchthat x<<y} is boundary of V(s).For any two disjoint fuzzy coalitions s and t, and two vectors x∈Rcar(s), y∈Rcar(t), we denote by (x, y) thevector in Rcar(s)∪car(t) whose car(s)-components are as in x and car(t)-components are as in y. The game (N, V) issuperadditive if for every two fuzzy coalitions s and t with car(s)∩car(t)=∅, and any two vectors x∈V(s), y∈V(t),we have (x, y)∈V(s∨t).The preimputation set of NTU fuzzy game (N, V) is I0=V0(N), imputation set is I=V0(N)∩R+N.3. BARGAINING SET FOR COOPERATIVE FUZZY GAMEWITH NON-TRANSFERABLEUTILITYWenbo Yang etc.[6] firstly extended the concept of bargaining set which was proposed by Davis-Machler[7]to TU fuzzy game. Here we will redefine the concept of bargaining set from the view point of Davis-Machler toNTU cooperative fuzzy game.Definition 3.1 Let x be an imputation of the NTU fuzzy game (N, V), then an objection of player k againstplayer l with respect to x is a pair (y, s), if siyi> sixi, for all i∈car(s), y∈V(s), where s∈N with car(s)≠0 and N,sk>0, sl=0, and y is a vector whose indices are the members of car(s).Definition 3.2 Let x be an imputation of the NTU fuzzy game (N, V), and let (y, s) be an objection ofplayer k against player l with respect to x. Then (z, t) is a counter objection of player l to the objection (y, s) ofplayer k, if)()(),()( tcarscariallforxysxztiiiiii∩∈−≥−)(\)(, scartcariallforxtztiiii∈≥)(tVz∈ ,where t∈ΦN with tk=0, tl>0, and ti≥si, for all i∈car(s)∩car(t), and z is a vector whose indices are the members ofcar(t).Definition 3.3 An imputation x belongs to the bargaining set MF{i}(V) for a NTU fuzzy cooperative game, iffor every objection, there exists a counter objection to it.Mos-Colell[8](1989) proposed another bargaining set, now we will redefine the concept to NTU fuzzy gameas follows.Definition 3.4 Let x be a preimputation of the NTU fuzzy game (N, V), for a fuzzy coalition s∈ΦN, ifThe Twelfth Wuhan International Conference on E-Business——Emerging Operations & Services Management Track 835car(s)≠0 and N, an objection on fuzzy coalition s with respect to x is a pair (y, s), where y∈Rcar(s) and satisfying:)(sVy∈ ,)(, scarixsysiiii∈≥ ,and at least one of the inequalities above is strict.Definition 3.5 Let x be a preimputation of the NTU fuzzy game (N, V), and let (y, s) be an objection tofuzzy coalition s with respect to x. Then (z, t) is a counter objection on fuzzy coalition t∈ΦN with respect to x,here ti≥si, and for all i∈car(s)∩car(t), z∈Rcar(t) satisfying)(tVz∈)(\)(,)()(),()(scartcariallforxtzttcarscariallforxysxztiiiiiiiiii∈≥∩∈−≥−and at least one of the inequalities above is strict.An objection is called justified, if there is no counter objection.Definition 3.6 A preimputation x belongs to the bargaining set MBF(V) of NTU fuzzy cooperative game ifthere is no justified objections with respect to x.Theorem 3.1 Let (N, V) be a supperadditive NTU fuzzy game, then)()(}{ VMBVMFiF⊆ .Proof. Suppose, for contradiction, that x∈MF{i}(V)\ MBF(V). Let (y, c) be a justified objection in the senseof MBF(V) at x. Chosen c such that c≥s, where s is an any fuzzy coalition which can cause a justified objectionin MBF(V) at x. Let ck>0, and ckyk>ckxk. From the concept of the bargaining set MBF(V), we have car(s)≠N. Letl∈N\car(c).By the comprehensiveness and nonlevelness of V(c), we can modify y, and y is still in V(c). Let thek-component of y will decrease, but is still great then xk, and the other components of y will increase. So, weobtain a vector )(~ cVy ∈ satisfyingkkkkxcyc >~}{\)(,~ kccariycyciiii∈>Clearly, ),~( cy is an objection of k against l at x. Since )(}{ VMx iF∈ , there exists a counter objection (z, d) to),~( cy satisfying )()(, ccardcaricdiiΙ∈≥ , 0=kd , 0>ld . So we have)()(),()~()( ccardcarixycxycxzdiiiiiiiiiΙ∈−>−≥− (1)                                                            )(\)(, ccardcarixdzdiiii∈≥                                                           (2)Thus, (z, d) satisfies almost all the requirements to be a counter objection to ),( cy in the sense of )(VMBF,except one condition, i.e., at least one strict inequality in (1) and (2) is true. Since ),( cy is justified, and in theinequality (1), )()( dcarccar Ι must be an empty set. But from the property of surperadditivity, we know that)),,(( dczy ∨ is a justified objection in the sense of )(VMBFat x , so )( dcc ∨< , contradicting ourassumption. The conclusion is true.836 The Twelfth Wuhan International Conference on E-Business——Emerging Operations & Services Management Track4. CONCLUSIONSNTU cooperative game can be converted to TU cooperative game under specific conditions. In the generalsense, NTU cooperative game can be viewed as the extension of TU cooperative game. But, many properties ofNTU cooperative game can help to study the properties of the TU cooperative games. In this paper we extendedtwo concepts of the bargaining set for TU cooperative game to NTU cooperative fuzzy game, and proved thatthe inclusion relation between these two bargaining sets is still satisfied. The problem of the bargaining sets howcan be used in economic management effectively is still open[10]. We will work on it in future.REFERENCES[1] Aubin, J.P.: Coeur et valeur des jeux flous a paiements lateraux, Comptes Rendus de l’ Academie des Sciences Paris 279A, 891-894(1974).[2] Aubin, J.P.: Cooperative fuzzy games, Mathematics of Operations 6, 1-13(1981)[3] Tijs, S., R. Branzei. S. Ishihara, and S. Muto: On cores and stable sets for fuzzy games, Fuzzy Sets and Systems 146,285-296(2004)[4] Muto, S., S. Ishihara, E. Fukuda, S. Tijs, and R. Branzei: Generalized cores and stable sets for fuzzy games, InternationalGame Theory Review 8, 95-109(2006)[5] Zhang Fengrong, Zhang Shengkai, Xie liyang, The Fuzzy NTU-game, Systems Engineering, Vol. 22, No. 10, Oct., 2004[6] Wenbo, Y., Jiuqiang L., Xiaodong L.: Aubin Cores and Bargaining Sets for Convex Cooperative Fuzzy Games,International Journal of Game Theory, Online first 2010[7] Aumann, R.J., Maschler, M.: The bargaining set for cooperative games. In: Dresher, M., Shapley, L.S., Tucker, A.W.(eds)Advances in game theory, Princeton University Press, Princeton, N.J., pp. 443-476(1964)[8] Mas-Colell, A.: An equivalence theorem for a bargaining set. J Math Econ 18, 129-138(1989)[9] Branzei, R., D. Dimitrov, and S. Tijs: Convex fuzzy games and participation monotonic allocation schemes, Fuzzy Setsand Systems 139, 267-281(2003)[10] Xie Shiyu, Economics Game Theory, Fudan university press,(1997)

A Situation of Economic Management in NTU Cooperative Fuzzy Games

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A Situation of Economic Management in NTU Cooperative Fuzzy Games

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